Integrand size = 32, antiderivative size = 334 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^8 \sqrt {a+b x^2}} \]
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Time = 0.34 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1817, 12, 277, 198, 197} \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt {a+b x^2}}+\frac {64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]
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Rule 12
Rule 197
Rule 198
Rule 277
Rule 1817
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {14 A b-7 a \left (B+C x^2+D x^4\right )}{x^6 \left (a+b x^2\right )^{9/2}} \, dx}{7 a} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {12 b (14 A b-7 a B)-5 a \left (-7 a C-7 a D x^2\right )}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{35 a^2} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {10 b \left (168 A b^2-84 a b B+35 a^2 C\right )-105 a^3 D}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{105 a^3} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac {\left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a^3} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {\left (8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^4} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {\left (16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^5} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {\left (64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^6} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {\left (128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^7} \\ & = -\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^8 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {6144 A b^7 x^{14}-3072 a b^6 x^{12} \left (-7 A+B x^2\right )+256 a^2 b^5 x^{10} \left (105 A-42 B x^2+5 C x^4\right )+14 a^6 b x^2 \left (3 A+6 B x^2+25 C x^4-60 D x^6\right )+112 a^4 b^3 x^6 \left (15 A-60 B x^2+50 C x^4-12 D x^6\right )+128 a^3 b^4 x^8 \left (105 A-105 B x^2+35 C x^4-3 D x^6\right )-56 a^5 b^2 x^4 \left (3 A+15 B x^2-50 C x^4+30 D x^6\right )-a^7 \left (15 A+21 B x^2+35 x^4 \left (C+3 D x^2\right )\right )}{105 a^8 x^7 \left (a+b x^2\right )^{7/2}} \]
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Time = 3.58 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {\left (-105 D x^{6}-35 C \,x^{4}-21 x^{2} B -15 A \right ) a^{7}+42 b \,x^{2} \left (-20 D x^{6}+\frac {25}{3} C \,x^{4}+2 x^{2} B +A \right ) a^{6}-168 b^{2} x^{4} \left (10 D x^{6}-\frac {50}{3} C \,x^{4}+5 x^{2} B +A \right ) a^{5}+1680 \left (-\frac {4}{5} D x^{6}+\frac {10}{3} C \,x^{4}-4 x^{2} B +A \right ) b^{3} x^{6} a^{4}+13440 \left (-\frac {1}{35} D x^{6}+\frac {1}{3} C \,x^{4}-x^{2} B +A \right ) b^{4} x^{8} a^{3}+26880 \left (\frac {1}{21} C \,x^{4}-\frac {2}{5} x^{2} B +A \right ) b^{5} x^{10} a^{2}+21504 b^{6} x^{12} \left (-\frac {x^{2} B}{7}+A \right ) a +6144 A \,b^{7} x^{14}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{7} a^{8}}\) | \(218\) |
gosper | \(-\frac {-6144 A \,b^{7} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}+384 D a^{3} b^{4} x^{14}-21504 A a \,b^{6} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}+1344 D a^{4} b^{3} x^{12}-26880 A \,a^{2} b^{5} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}+1680 D a^{5} b^{2} x^{10}-13440 A \,a^{3} b^{4} x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}+840 D a^{6} b \,x^{8}-1680 A \,a^{4} b^{3} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+105 D a^{7} x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8}}\) | \(301\) |
trager | \(-\frac {-6144 A \,b^{7} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}+384 D a^{3} b^{4} x^{14}-21504 A a \,b^{6} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}+1344 D a^{4} b^{3} x^{12}-26880 A \,a^{2} b^{5} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}+1680 D a^{5} b^{2} x^{10}-13440 A \,a^{3} b^{4} x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}+840 D a^{6} b \,x^{8}-1680 A \,a^{4} b^{3} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+105 D a^{7} x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8}}\) | \(301\) |
default | \(A \left (-\frac {1}{7 a \,x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 b \left (-\frac {1}{5 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {12 b \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\right )}{a}\right )+C \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )+D \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )+B \left (-\frac {1}{5 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {12 b \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\right )\) | \(542\) |
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Time = 0.70 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left (128 \, {\left (3 \, D a^{3} b^{4} - 10 \, C a^{2} b^{5} + 24 \, B a b^{6} - 48 \, A b^{7}\right )} x^{14} + 448 \, {\left (3 \, D a^{4} b^{3} - 10 \, C a^{3} b^{4} + 24 \, B a^{2} b^{5} - 48 \, A a b^{6}\right )} x^{12} + 560 \, {\left (3 \, D a^{5} b^{2} - 10 \, C a^{4} b^{3} + 24 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5}\right )} x^{10} + 280 \, {\left (3 \, D a^{6} b - 10 \, C a^{5} b^{2} + 24 \, B a^{4} b^{3} - 48 \, A a^{3} b^{4}\right )} x^{8} + 15 \, A a^{7} + 35 \, {\left (3 \, D a^{7} - 10 \, C a^{6} b + 24 \, B a^{5} b^{2} - 48 \, A a^{4} b^{3}\right )} x^{6} + 7 \, {\left (5 \, C a^{7} - 12 \, B a^{6} b + 24 \, A a^{5} b^{2}\right )} x^{4} + 21 \, {\left (B a^{7} - 2 \, A a^{6} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{8} b^{4} x^{15} + 4 \, a^{9} b^{3} x^{13} + 6 \, a^{10} b^{2} x^{11} + 4 \, a^{11} b x^{9} + a^{12} x^{7}\right )}} \]
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Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.46 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {128 \, D b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, D b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, D b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, D b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, C b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, C b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, C b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, C b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, B b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, B b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, B b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, B b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} + \frac {2048 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{8}} + \frac {1024 \, A b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{7}} + \frac {768 \, A b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{6}} + \frac {128 \, A b^{4} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} - \frac {D}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, C b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, B b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} + \frac {16 \, A b^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x} - \frac {C}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, B b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {8 \, A b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x^{3}} - \frac {B}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} + \frac {2 \, A b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{5}} - \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (300) = 600\).
Time = 0.32 (sec) , antiderivative size = 938, normalized size of antiderivative = 2.81 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {{\left (279 \, D a^{21} b^{7} - 790 \, C a^{20} b^{8} + 1686 \, B a^{19} b^{9} - 3072 \, A a^{18} b^{10}\right )} x^{2}}{a^{26} b^{3}} + \frac {7 \, {\left (132 \, D a^{22} b^{6} - 365 \, C a^{21} b^{7} + 768 \, B a^{20} b^{8} - 1386 \, A a^{19} b^{9}\right )}}{a^{26} b^{3}}\right )} + \frac {35 \, {\left (30 \, D a^{23} b^{5} - 80 \, C a^{22} b^{6} + 165 \, B a^{21} b^{7} - 294 \, A a^{20} b^{8}\right )}}{a^{26} b^{3}}\right )} x^{2} + \frac {105 \, {\left (4 \, D a^{24} b^{4} - 10 \, C a^{23} b^{5} + 20 \, B a^{22} b^{6} - 35 \, A a^{21} b^{7}\right )}}{a^{26} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} D a^{3} \sqrt {b} - 420 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} C a^{2} b^{\frac {3}{2}} + 1050 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {5}{2}} - 2100 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {7}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} D a^{4} \sqrt {b} + 2730 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} C a^{3} b^{\frac {3}{2}} - 7140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {5}{2}} + 14700 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a b^{\frac {7}{2}} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} D a^{5} \sqrt {b} - 7210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} C a^{4} b^{\frac {3}{2}} + 19950 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {5}{2}} - 42840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {7}{2}} - 2100 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} D a^{6} \sqrt {b} + 9940 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{5} b^{\frac {3}{2}} - 28560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {5}{2}} + 64680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{3} b^{\frac {7}{2}} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} D a^{7} \sqrt {b} - 7560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{6} b^{\frac {3}{2}} + 21966 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {5}{2}} - 49812 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {7}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} D a^{8} \sqrt {b} + 3010 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{7} b^{\frac {3}{2}} - 8652 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {5}{2}} + 19404 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{5} b^{\frac {7}{2}} + 105 \, D a^{9} \sqrt {b} - 490 \, C a^{8} b^{\frac {3}{2}} + 1386 \, B a^{7} b^{\frac {5}{2}} - 3072 \, A a^{6} b^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{7}} \]
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Time = 7.74 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {61\,B\,b}{35\,a^3}+\frac {78\,B\,b^2\,x^2}{35\,a^4}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {128\,C\,b}{21\,a^5}+\frac {256\,C\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {C}{3\,a^2}+\frac {19\,C\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {167\,A\,b^2}{35\,a^4}+\frac {191\,A\,b^3\,x^2}{35\,a^5}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {1024\,A\,b^3}{35\,a^7}+\frac {2048\,A\,b^4\,x^2}{35\,a^8}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {512\,B\,b^2}{35\,a^6}+\frac {1024\,B\,b^3\,x^2}{35\,a^7}}{x\,\sqrt {b\,x^2+a}}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a^5\,x^7}-\frac {B\,\sqrt {b\,x^2+a}}{5\,a^5\,x^5}-\frac {{\left (\frac {a}{b\,x^2}+1\right )}^{9/2}\,D\,{{}}_2{\mathrm {F}}_1\left (\frac {9}{2},5;\ 6;\ -\frac {a}{b\,x^2}\right )}{10\,x\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {34\,A\,b\,\sqrt {b\,x^2+a}}{35\,a^6\,x^5}-\frac {B\,b}{7\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {32\,C\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {C\,b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {58\,A\,b^3}{7\,a^6\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {A\,b^2}{7\,a^3\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {27\,B\,b^2}{7\,a^5\,x\,{\left (b\,x^2+a\right )}^{3/2}} \]
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